sri s.ramasamy naidu memorial college...

SRNMC Regulation-2016 Syllabus

SRI S.RAMASAMY NAIDU MEMORIAL COLLEGE

SATTUR- 626 203 (An Autonomous institution affiliated to the Madurai Kamaraj University, Madurai)

(Re-Accredited with Grade ‘A’ by NAAC)

M.Sc. DEGREE COURSE IN

MATHEMATICS

SYLLABUS AND REGULATIONS

UNDER

CHOICE BASED CREDIT SYSTEM (CBCS)

(Those who are joining in 2016-2017 and after)


Objectives

The syllabus for Mathematics has been designed in such a way that the

students, whenthey go out, will be capable of facing the competitive

situation prevailing now. The focus is on getting placement for all the

students.

Eligibility

A candidate with a pass in B.Sc Mathematics degree conducted by

the Madurai Kamaraj University or any other university duly

recognized by the syndicate of Madurai Kamaraj University as

equivalent thereto is eligible to join the course.

Evaluation

The performance of the student is evaluated in terms of percentage of marks with a

provision for conversion to grade points. Evaluation for each course shall be done by a

continuous internal assessment by the concerned course teachers as well as by an end semester

examination which will be a written type examination of 3 hours duration. The ratio of marks

to be allotted to continous internal assessment and to end semester examination is 25:75( i.e

internal 25 marks and external 75 marks for theory), Practical 40 : 60.

The components for continuous internal assessment are:

Theory:

Two tests and their average ---15 marks

Seminar/Group Discussion --- 5 marks

Assignment --- 5 marks

Total --- 25 marks

Practical:

Two tests and their average ---20 marks

Record --- 5 marks

Observation Evaluation ---15 marks

Total --- 40 marks


Passing Minimum

A) Theory:

1. No separate pass minimum for internal

2. 45% is the pass minimum for the External

3. 50% of the aggregate (external+ internal)

B) Practical:




C) Project:




4. Minimum of 25 pages in the project work excluding

1. Introduction

2. Reference

3. Bibliography

4. Tables

5. Graphs

Passing Minimum

A candidate passes the P.G. program by scoring a minimum of 50%

(internal + external) in each paper of the course. No minimum mark for internal

assessment. External minimum for external assessment is 45% and external

minimum is 34 out of 75.

Duration of the Course

The duration of the course shall be two academic years comprising four semesters.


Credits

The term „credit‟ refers to the weightage given to the course, usually in relation to the instructional

hours assigned to it. The total credits required for completing Master of Science in Mathematics is 90. The

particulars of the credits for individual components and the courses are placed on Table-1

Extra Credits Paper:

1. This paper is optional. Students may or may not select this paper. If he/she selects this paper

and if he/she passes the paper, then 3 extra credits will be added in his/her total credit to the

degree, even otherwise, it won‟t affect the completion of degree.

2. Though this paper is common to all PG programmes, the syllabus varies according to the

subject selected by the department.

3. The title of this paper is “Model Paper for NET Examinations”

4. Examination for this paper will be held at the end of the 4th

semester examinations.

5. There is no internal examination and only external examination for this paper.

6. Maximum marks for this paper is 100.

Question Type of

Question

No.of

Question

No.of

Question to

be answered

Marks for

Each

question

Total

Marks

A.Q.No(1-5)

Q.No(6-10)

Multiple

choice

(one from

each unit)

5 5 1

10

Sentence from

(one from

each unit)

5 5 1

B.Q.No(11-

15)

Either or type

(one from

each unit)

5 5 7 35

C.Q.No(16-

20)

Open choice

(One from

each unit)

5 3 10 30


M.Sc., (Mathematics)

Table-1: Course Pattern

(For those who are joining in the academic year 2016-2017 and after)

Semester I II III IV V Total

Hours

Total

Credits

No. of

Courses

I Core

6(5)

Core

6(5)

Core

6(4)

Core

6(4)

Elective

6(4)

30 22 5

II Core

6(5)

Core

6(5)

Core

6(4)

Core

6(4)

Elective

6(4)

30 22 5

III Core

6(5)

Core

6(5)

Core

6(5)

Core

6(4)

Elective

6(4)

30 23 5

IV Core

6(5)

Core

6(5)

Core

6(5)

Elective

6(4)

Project

6(4)

30 23 5

Total

120 90 20

Extra Credits

paper

3

Grand Total 93


M.Sc., (Mathematics) Table-1: Course Details and Scheme of Examination

(For those who are joining in the academic year 2016-2017 and after)

First Semester

S.

No

Title of the

Paper

Subject

Code Hours /

week Credits

Exam

hrs

Marks

Total

Marks Int Ext

1. Groups and

Rings

P16MAC11 6 5 3 25 75 100

2. Mathematical

Analysis

P16MAC12 6 5 3 25 75 100

3. Differential

Geometry

P16MAC13 6 4 3 25 75 100

4. Mathematical

Statistics

P16MAC14 6 4 3 25 75 100

5.

Elective

a) Classical

Mechanics

b)Graph

Theory

P16MAE11

P16MAE12

6 4 3 25 75 100


SECOND SEMESTER

S.

No

Title of the

Paper

Subject

Code

Hours/

week Credits

Exam

hrs

Marks Total

Marks Int Ext

1. Linear Algebra P16MAC21 6 5 3 25 75 100

2. Advanced

Mathematical

Analysis

P16MAC22

6 5 3 25 75 100

3. Differential

Equations

P16MAC23 6 4 3 25 75 100

4. Advanced

Mathematical

Statistics

P16MAC24

6 4 3 25 75 100

Elective

a)Combinatorial

Mathematics

b) Fuzzy

Algebra

P16MAE21

P16MAE22

6 4 3 25 75 100


THIRD SEMESTER

S.

No

Title of the

Paper

Subject

Code Hours

per

week

Credits Exam

hrs

Marks

Total

Marks Int Ext

1. Galois Theory

and Lattices

P16MAC31 6 5 3 25 75 100

2. Measure and

Integration

P16MAC32 6 5 3 25 75 100

3. Topology P16MAC33 6 5 3 25 75 100

4. Stochastic

Processes

P16MAC34 6 4 3 40 75 100

Major

Elective

a) Numerical

Analysis

b) Integral

Equations

P16MAE31

P16MAE32

6

4

3

25

75

100


FOURTH SEMESTER

S. No Title of the

Paper

Subject

Code

Hours

per

week

Credits Exam

hrs

Marks Total

Marks Int Ext

1. Complex

Analysis

P16MAC41 6 5 3 25 75 100

2. Functional

Analysis

P16MAC42 6 5 3 25 75 100

3. Operations

Research

P16MAC43 6 5 3

25

75 100

4. Major Elective

a) Advanced

Topology

b) Number

Theory and

Cryptography

P16MAE41

P16MAE42

6

4

3

25

75

100

Project P16MAPT41 6 4 3 25 75 100

Extra Credits paper

Model paper for

NET/SET

Examination

P16MAX41 3 3



(An Autonomous Institution Re-accredited with „A‟ Grade by NAAC)

SATTUR – 626 203. Department of Mathematics

(For those who are joining in 2016-2017 and after)

SYLLABUS

Programme : M. Sc., Mathematics Subject Code :P16MAC11

Semester : I No. of Hours allotted : 6 / Week

Paper : Core - Paper I No. of Credits : 5

Title of the Paper - GROUPS AND RINGS

Objectives:

To know more concepts in algebra which helps them to develop thinking and improve

mathematical ability.

To understand the basic algebraic structures prevailing in the set of numbers. Unit I

Counting Principle – Conjugacy – Cauchy‟s Theorem - Sylow‟s Theorem – Second Part

Part of Sylow‟s Theorem – Third part of Sylow‟s Theorem.

Unit II

Direct Products - Finite Abelian Groups –Invariants.

Unit III

Ideals and Quotient Rings- More Ideals and Quotient Rings -The Field of Quotients of

an Integral Domain.

Unit IV

Euclidean Rings - A particular Euclidean Ring.

Unit V

Polynomial Rings - Polynomials over the Rational Fields - Polynomial Rings over

Commutative rings.

Text Book:

Title of the book : Topics in Algebra

Name of the Author : I.N. Herstein

Publisher

Edition/Year

;John Wiley and Sons

:Second Edition,1999, Third reprint 2008


Unit I :

Chapter 2: Sections 2.11, 2.12.

Unit II:


Unit III:

Chapter 3: Sections 3.4, 3.5, 3.6.

Unit IV:


Unit V:

Chapter 3: Sections 3.9, 3.10 and 3.11.

Reference Books:

Reference Book - 1:

Title of the book : University Algebra

Name of the Author : N.S.Goplakrishnan

Publisher

Edition/Year

Reference Book - 2

Title of the book

: New Age Inernational Pvt Ltd, Newdelhi

: Second Edition

: Basic Abstract Algebra

Name of the Author : P.B. Bhattacharya

Publisher

Edition/Year

: S.K. Jain, S.R. Nagpaul Cambridge University Press

: 1995 Reprinted 2009

Prepared by : Dr. V.Thiripurasundari

Signature :


SRI S.RAMASAMY NAIDU MEMORIAL COLLEGE (An Autonomous Institution Re-accredited with „A‟ Grade by NAAC)

SATTUR – 626 203.

Department of Mathematics


SYLLABUS


Semester : I No.of Hours allotted : 6 / Week

Paper : Core - Paper II No.of Credits : 5

Title of the Paper - MATHEMATICAL ANALYSIS

Objectives:

To give a comprehensive idea about the underlying principles of real analysis.

To enable the students to have a good foundation in all the concepts in sequences

and series.

Unit I

Finite, countable and uncountable sets - Metric spaces.

Unit II

Compact sets, Perfect Sets and Connected Sets. Convergent sequences, subsequences

and Cauchy sequences - Upper and lower limits - Some special sequences.

Unit III

Series - Series of Non-negative Terms - The Number e. The Root and Ratio Tests -

Power Series - Summation by parts - absolute convergence - addition and multiplication of

series.

Unit IV

Limits of function - Continuous Functions - Continuity and Compactness - continuity

and Connectedness – Discontinuities - monotonic functions - infinite limits and limits at

infinity.

Unit V

The derivative of Real function - Mean Value Theorems - The continuity of

derivatives - L‟Hospital‟s Rule -Derivatives of Higher Order - Taylor‟s Theorem,

Differentiation of vector-valued Functions.


Text Book:

Title of the book

: Principles of Mathematical Analysis

Name of the Author : Walter Rudin

Publisher

Edition/Year

Unit I :

: McGraw Hill, International student edition

: Third Edition,1976

Chapter 2: Sections 2.1 to 2.30

Unit II:

Chapter 2: Sections 2.31 to 2.47.


Unit III:


Unit IV:


Unit V :


Reference Book:

Reference Book - 1

Title of the book : Mathematical Analysis

Name of the Author : Tom M.Apostal

Publisher

Edition/Year

Reference Book - 2

Title of the book

: Addision-Wesley Pub Company

: Second Edition, 1978

: Real Analysis

Name of the Author : N.L.Carothers

Publisher

Edition/Year

: Cambridge University Press

: First South Asian Edition 2006

Prepared by : Dr. S.Rajaram

Signature :




SATTUR – 626 203.

Department of Mathematics (For those who are joining in 2016-2017 and after)

SYLLABUS



Paper : Core - Paper III No.of Credits : 4

Title of the Paper - DIFFERENTIAL GEOMETRY

Objectives:

To get clear idea on three dimensional surfaces.

To get new ideas and techniques which play a prominent role in current research in global differential geometry.

Unit I

Introductory remarks about space curves, definition, arc length, tangent, normal and

binormal - Curvature and torsion of a curve given as the intersection of two surfaces -

Contact between curves and surfaces - Tangent surface - Involutes and Evolutes.

Unit II

Intrinsic equations - Fundamental existence theorem for space curves – Helics -

Definition of a surface, curves on surface - Surfaces of revolution - Helicoids.

Unit III

Metric - Direction Coefficients - Family of curves - Isometric correspondence-

Intrinsic Properties – Geodesics - Canonical geodesic equations - Normal property of

geodesics.

Unit IV

Geodesic Curvature - Gauss Bonnet Theorem - Gaussian curvature – Surface of

constant curvature – Minding Theorem.

Unit V

The Second fundamental form - Principal curvatures - Lines of curvature –

Developables - Developables associated with space curves - Developables associated with

curves on surfaces - Minimal surfaces - Ruled surfaces.


Text Book:

1.Title of the book : An introduction to differential geometry

Name of the Authos : T.G.Willmore

Publisher : Oxford University Press

Edition/Year : 1959, Twenty third impression 2008

Unit I :

Chapter 1: Sections 1 to 7

Unit II :

Chapter 1: Section 8 & 9.

Chapter 2 : Sections 1 to 4

Unit III:

Chapter 2: Sections 5 to 12.

Unit IV :

Chapter 2: Section 15 to 18.

Unit V:


Reference Book:

Reference Book - 1

Title of the book : Differential Geometry of Three Dimensions

Name of the Author : Charles Emest Weatherburn

Publisher : The University Press

Edition/Year : 1998

Reference Book - 2

Title of the book : Differential Geometry : A first Course

Name of the Author : D.Somasundaram

Publisher :Narosa Publishing House

Edition/Year : 2006

Prepared by : Dr. S. Murugesan

Signature :




SATTUR – 626 203.

Department of Mathematics (For those who are joining in 2016-2017 and after)

SYLLABUS



Paper : Core - Paper IV No.of Credits : 4

Title of the Paper – MATHEMATICAL STATISTICS

Objectives:

To develop the ability in the students to understand more concepts in Statistics.

To know more about various distributions.

Unit I

The probability set functions, Conditional probability and independence, random

variables of the discrete type, random variables of the continuous type, properties of the

distribution function, expectation of a random variable, some special expectations, Chebyshev‟s

inequality.

Unit II

Distributions of two random variables, conditional distributions and expectations, the

correlation coefficient, independent random variables, extension to several random variables.

Unit III

The binomial and related distributions, Poisson distribution, The gamma and Chi-

square distributions, the normal distribution, the Bivariate normal distribution.

Unit IV

Sampling theory, transformations of variables of the discrete type, transformations of

variables of the continuous type, the Beta, t, F distributions, extensions of the change-of-

variable technique, distributions of order statistics, The moment generating function

technique, the distributions of X and nS2 /σ

2, expectations of functions of random variables.

Unit V

Convergence in distribution, convergence in probability, limiting moment generating

functions, the central limit theorem, some theorems on limiting distributions.


Text Book:

Title of the book : Introduction to Mathematical Statistics

Names of the Authors : R.V.Hogg and A.T.Craig

Publisher : Pearson Education, Asia

Edition/year : V Edition, 2002

Unit I:

Chapter 1: Sections : 1.3 to 1.10.

Unit II

Chapter 2: Sections: 2.1 to 2.5.

Unit III


Unit IV


Unit V


Reference Book:

Reference Book - 1

Title of the book : An Introductory Statistics

Name of the Author : Ross, Sheldon.M

Publisher : USA Academic Press

Edition/Year : 2006

Reference Book – 2

Title of the book : Introduction to Probability Theory and its Applications

Name of the Author : William Feller

Publisher : Wiley India

Edition/Year: : 3rd

Edition Volume I /2011 Prepared by : Dr. N. Soundararaj

Signature :




SATTUR – 626 203.



SYLLABUS

Programme : M. Sc., Mathematics Subject Code :P16MAE11


Paper : Paper V Elective I (a) No.of Credits : 4

Title of the Paper - CLASSICAL MECHANICS

Objectives:

To promote logical thinking to understand the basic principles

of mechanics to be applied to practical problems.

To provide strong foundation about elementary principles and variational

Principles of mechanics

Unit I

Mechanics of a particle, Mechanics of a system of particles, Constraints.

Unit II

D‟Alembert‟s principle and Lagrange‟s equations, velocity dependent potentials and

the dissipation function, Hamilton‟s principle, some techniques of the calculus of variations.

Unit III

Derivation of Lagrange‟s equations from Hamilton‟s principle, Extension of

Hamilton‟s principle to nonholonomic systems, Advantage of a variational principle

formulation, Conservation theorems and Symmetry properties.

Unit IV

Reduction to the equivalent one – body problem, the equations of motion and first

integrals. The equivalent one – dimensional problem and classification of orbits, the Virial

theorem.

Unit V

The differential equation for the orbit and integrable power-law potentials, conditions

for closed orbits (Bertrand‟s theorem), The Kepler problem: Inverse square law of force, The

motion in time in the Kepler problem, The Laplace – Runge- Lenz vector.


Text Book:

Title of the book

: Classical Mechanics

Name of the Author : H.Goldstein

Publisher

Edition/Year

Unit I :

: Addison Wesley, New York

: Second edition, 1980

Chapter 1: Sections 1.1 – 1.3

Unit II :

Chapter 1: Sections 1.4 - 1.5.


Unit III:


Unit IV :

`Chapter 3: Sections 3.1 – 3.4

Unit V:

Chapter 3: Sections 3.5 – 3.9.

Reference Books

Reference Book - 1

Title of the book : Classical Mechanics

Name of the Author : D.T.Greenwood,

Publisher

Edition/Year

Reference Book - 2

: Prentice Hall of India Pvt. ltd,NewDelhi

: 1979

Title of the book : Classical Mechanics

Name of the Author : D.E.Rutherford

Publisher : Oliver and Boyd

Edition/Year : 1987

Prepared by : Dr.N.Soundararaj

Signature :






SYLLABUS



Paper : Paper V - Elective I(b) No.of Credits : 4

Title of the Paper - GRAPH THEORY

Objectives:

To motivate the students about the fundamental principles of graph theory.

To give clear idea about the terms and definitions and problems of graph theory.

Unit I

Graphs and simple graphs, graph isomorphism, the incidence and adjacency matrices,

Subgraphs, Vertex degrees, Paths and Connection, Cycles, the shortest path problem,

Sperner‟s lemma.

Unit II

Trees, cut edges and Bonds, cut vertices, cayley‟s formula, The connector problem,

connectivity, blocks, construction of reliable communication networks.

Unit III

Euler tours, Hamilton cycles, the Chinese postman problem, the travelling salesman

problem.

Unit IV

Matchings, Matchings and coverings in bipartite graphs, Perfect matchings. The

personnel assignment problem, the optimal assignment problem.

Unit V

Edge chromatic number, Vizing‟s theorem, the timetabling problem.


Text Book:

Title of the book

: Graph theory with Applications

Names of the authors : J.A. Bondy and U.S.R.Murty

Publisher

Edition/Year

Unit I :

: North Holand

: 5thprinting /1983


Unit II :



Unit III:


Unit IV :


Unit V:


Reference Book:

Reference Book - 1

Title of the book : Graph Theory with applications to Engineering and Computer Science

Name of the Author : Narasing Deo,

Publisher

Edition/Year


Title of the book

: Pretice Hall of India(P) Ltd,NewDelhi

: 2007

: Introduction to Graph Theroy

Name of the Author : Douglas B.West

Publisher

Edition/Year

: Pearson Education Ltd

: 2nd

/ 2002

Prepared by : Dr. R. Sridevi

Signature :




SATTUR – 626 203.

Department of Mathematics (For those who are joining in 2016 -2017 and after)

SYLLABUS


Semester : II No. of Hours allotted : 6 / Week

Paper : Core - Paper VI No. of Credits : 5

Title of the Paper- LINEAR ALGEBRA

Objectives:

To know more concepts in algebra which helps them to develop thinking and

improve mathematical ability.

To study concrete examples and to do problems.

Unit I

Dual spaces – Annihilator - Inner Product spaces – Schwarz inequality – Orthogonal –

Orthonormal set – Gram-Schmidt Orthogonalization Process - Modules – Submodules.

Unit II

The algebra of linear transformations – Regular – Singular – Range - Characteristic roots

- Matrix of linear transformations.

Unit III

Canonical forms - Triangular form -Nilpotent transformations – Index of Nilpotence – A decomposition of V: Jordan form.

Unit IV

Rational Canonical form - Trace and Transpose.

Unit V

Determinants – Cramer‟s Rule – Secular equation - Hermitian - Unitary and Normal

Transformations.


Text Book:

Title of the book

: Topics in Algebra

Name of the author : I.N.Herstein

Publisher

Edition/Year

Unit I :

: John Wiley and Sons

: Second Edition 1999 Reprint 2008


Unit II :


Unit III:


Unit IV :


Unit V:

Chapter 6: Sections 6.9 and 6.10.

Reference Books:

Reference Book - 1

Title of the book : A first Course in Algebra

Name of the Author: J.B. Fraleigh

Publisher

Edition/Year

Reference Book - 2

: Addition –Wiely Longman Znc.Reading, Massachuetts

: 1991

Title of the book : Basic Abstract Algebra

Name of the Author: P.M. Bhattacharya

Publisher

Edition/Year

: S.K. Jain, S.R. Nagpaul Cambridge University Press

: Second Edition, 1995, Reprinted 2009

Prepared by : Dr. V.Thiripurasundari

Signature :



SATTUR – 626 203.


(For those who are joining in 2016 -2017 and after)

SYLLABUS


Semester : II No.of Hours allotted : 6 / Week

Paper : Core - Paper VII No.of Credits : 5

Title of the Paper - ADVANCED MATHEMATICAL ANALYSIS Objectives:

To develop the firm footing in Analysis.

To introduce Riemann integration.

Unit I

Definitions and existence of the integral - Properties of the Integral - Integration and

Differentiation - Integration of vector valued functions - Rectifiable curves.

Unit II

Uniform convergence - uniform convergence and continuity - uniform convergence and

differentiation - equicontinuous families of functions -The Weierstrass theorem.

Unit III

Power series - The exponential and logarithmic functions - The trigonometric

Functions - The algebraic completeness of the Complex field - Fourier series - The Gamma

function.

Unit IV

Linear Transformations – Differentiation – The contraction principle – The inverse

function theorem.

Unit V

The implicit function theorem – Determinants- Jacobians


Text Book:

Title of the book

: Principles of Mathematical Analysis

Name of the author : Walter Rudin

Publisher

Edition/Year

Unit I :

: McGraw Hill

: Third Edition, International Student Edition 1976


Unit II :


Unit III:


Unit IV :


Unit V:

Chapter 9: Sections 9.26 – 9.29, 9.33 to 9.38.

Reference Book:

Reference Book - 1

Title of the book : Mathematical Analysis

Name of the Author : Jom M.Apostal

Publisher

Edition/Year

Reference Book - 2

Title of the book

: Addision-Wesley Pub. Company

: Second Edition, 1978.

: Real Analysis

Name of the Author : N.L.Carothers

Publisher

Edition/Year


: First South Asian Edition 2006

Prepared by : Dr.S.Rajaram

Signature :




SATTUR – 626 203.



SYLLABUS



Paper : Core - Paper VIII No.of Credits : 4

Title of the Paper- DIFFERENTIAL EQUATIONS

Objectives:

To introduce the methods of solving ordinary differential equations.

To introduce the methods of solving partial differential equations.

Unit I

Introduction - Initial value problems for the homogeneous equation - Solutions of the

homogeneous equation -The Wronskian and linear independence - Reduction of the order of

a homogeneous equations with analytic coefficients - The Legendre equation.

Unit II

Introduction - The Euler equation - Second order equations with regular singular

points- example -Second order equations with regular singular points-The general case - The

Bessel equation, The Bessel equation (continued).

Unit III

Introduction- Equations with variable separated - Exact equations - The method of

successive approximations - The Lipschitz condition - Convergence of the successive

approximations- Approximations to and Uniqueness of, solutions.

Unit IV

Partial differential equations – origins of First-order Partial Differential Equations –

Linear equations of the first order – Integral Surfaces Passing Through a given curve –

Surfaces orthogonal to a given system of surfaces.

Unit V

Nonlinear Partial Differential Equations of the First Order – Compatible systems of

First-order Equations – Charpit‟s method – Special Types of first-order Equations.


Text Books:

Text Book - 1

Title of the book : An introduction to ordinary differential equations

Name of the author : E.A. Coddington

Publisher : Prentice Hall of India

Edition/Year : 2010

Text Book - 2

Title of the book : Elements of Partial Differential Equations

Name of the author : I.N. Sneddon

Publisher : Tata McGraw Hill Book Company

Edition/Year : 1988

Unit I : (From Textbook-1)


Unit II : (From Textbook -1)

Chapter 4: Sections 1 to 4,7,8.

Unit III: (From Textbook-1)

Chapter 5: Sections 1 to 6, 8.

Unit IV: (From Textbook -2)

Chapter 2: Sections 2.1, 2.2, 2.4 to 2.6

Unit V: (From Textbook -2)

Chapter 2: Sections 2.7, 2.9 to 2.11.

Reference Books:

Reference Book - 1

Title of the book : Differential Equations

Name of the Author : G.F.Simmons

Publisher : Tata McGraw-Hill Education

Edition/Year : 01-May-2006

Reference Book - 2

Title of the book : An Introduction to partial differential equation

Name of the Author : Yehuda Pinchover and Jacob Rubinstein

Publisher : Cambridge University press

Edition/Year : 2005

Prepared by :Dr.K.Nagarajan

Signature :




SATTUR – 626 203.


SYLLABUS



Paper : Core - Paper IX No.of Credits : 4

Title of the Paper – ADVANCED MATHEMATICAL STATISTICS Objectives:

To develop the skills in students to apply statistical methods to real problems.

To understand more concepts in statistics and to test hypothesis of different types.

Unit I

Point estimation, confidence intervals for means, confidence intervals for differences

of means, tests of statistical hypotheses.

Unit II

Measures of quality of estimators, a sufficient statistics for a parameter, properties of

a sufficient statistic, Completeness and uniqueness, the exponential class of probability

density functions, functions of a parameter. .

Unit III

Bayesian estimation, Fisher information and the Rao-Cramer inequality, Limiting

distributions of maximum likelihood estimators.

Unit IV

Certain best tests, uniformly most powerful tests, likelihood ratio tests.

Unit V

Distributions of certain quadratic forms, A test of equality of several means, non-

Central 2 and non-central F, Multiple comparisons, The analysis of variance.


Text Book:

Title of the book

: Introduction to Mathematical Statistics

Name of the author : R.V.Hogg and A.T.Craig

Publisher

Edition/Year

UNIT I :

: Pearson Education, Asia

: V Edition , 2002


UNIT II:


UNIT III:


UNIT IV:


UNIT V:


Reference Book:

Reference Book - 1

Title of the book : An Introductory Statistics

Name of the Author : Ross,Sheldom.M

Publisher

Edition/Year

Reference Book - 2

Title of the book

: USA, Academic Press

: 2006

: Introduction to Probability Theory and its Applications

Name of the Author : William Feller

Publisher

Edition/Year

: Wiley India

: 3rd

Edition Volume I /2011

Prepared by : Dr. N. Soundararaj

Signature :



SATTUR – 626 203.



SYLLABUS



Paper : Paper X - Elective II(a) No.of Credits : 4

Title of the Paper - COMBINATORIAL MATHEMATICS

Objectives:

To introduce the calculating capacity by dealing with enumerating problems.

To know about various application of mathematics in practical situations.

Unit I

Introduction – the rules of sum and product – permutations – combinations –

distribution of distinct objects – Distribution of non distinct objects.

Unit II

Introduction – generating functions for combinations – enumerators for permutations

– distributions of distinct objects into non distinct cells – partitions of integers – elementary

relations.

Unit III

Introduction – Linear recurrence relations with constant coefficients – Solution by the

technique of generating functions – Recurrence relations with two indices.

Unit IV

Introduction – The principle of inclusion and exclusion – The general formula –

Derangements – Permutations with Restrictions on relative positions.

Unit V

Introduction – Equivalence classes under permutation Groups – Equivalence classes

of functions – Weights and inventories of functions – Polya‟s fundamental theorem –

Generalization of Polya‟s Theorem.


Text Book:

Title of the book

: Introduction to Combinatorial Mathematics

Name of the author : C.L. Liu

Publisher

Edition/Year

Unit I :

: McGraw Hill

: 1968


Unit II :

Chapter 2: Sections 2.1 – 2.5 and 2.7.

Unit III:

Chapter 3: Sections 3.1,3.2, 3.3 and 3.5.

Unit IV :

Chapter 4: Sections 4.1 – 4.5 .

Unit V:

Chapter 5: Sections 5.1, 5.3 – 5.7.

Reference Book:

Reference Book - 1

Title of the book : A first course in Combinatorial Mathematics

Name of the Author : Ian Anderson

Publisher

Edition/Year

Reference Book - 2

Title of the book

: Oxford University Press

: 2005

: A Course in Combinatorics

Name of the Author : J.H.Van Lint,R.M.Wilson

Publisher

Edition/Year


: First Southasian Edition 2002

Prepared by : Dr. K.Nagarajan

Signature :




SATTUR – 626 203.


SYLLABUS



Paper : Paper X Elective II(b) No.of Credits : 4

Title of the Paper- FUZZY ALGEBRA

Objectives:

To impart the students with fundamentals of Fuzzy set theory and its applications.

To inculcate the habit of viewing objects as graded ones.

Unit I

Fuzzy sets – Basic types – Fuzzy sets – Basic concepts – Additional properties of α –

cuts – Representation of fuzzy sets – Extension principle for fuzzy sets – Types of operations

– Fuzzy complements.

Unit II

Fuzzy numbers – Linguistic variables – arithmetic operations on intervals – arithmetic

operation on fuzzy numbers, fuzzy equations.

Unit III

Fuzzy relation – Crisp versus fuzzy relations – projections and cyclinderic Extensions

– Binary fuzzy relations on a single set – Fuzzy equivalence relations- Fuzzy compatibility

relations – Fuzzy ordering relations, fuzzy morphisms.

Unit IV

Definition of Fuzzy Subgroups – Examples and Properties – Union of two fuzzy

subgroups – Fuzzy subgroups generated by a Fuzzy subsets – Fuzzy Normal Subgroups.

Unit V

Fuzzy normal subgroups under homomorphisms – Characteristics subgroups –Fuzzy

conjugate subgroups – Fuzzy Sylow subgroups


Text Books:

Text Book-1

Title of the book

: Fuzzy sets and Fuzzy logic – Theory and applications

Name of the Author : George J.Klir and B.Yuan

Publisher : Second edition, 2008

Edition/Year : Prentice Hall of India

Text Book-2

Title of the Book : Fuzzy Algebra Vol I

Name of the Author : Rajeshkumar

Publisher

Edition/ Year

: University of Delhi, Publication Division

: 1993

Unit I : (From Text Book – 1)

Chapter 1: (1.2 to 1.4)

Chapter 2: (2.1 to 2.3)

Unit II : (From Text Book – 1)

Chapter 3: (3.1, 3.2)

Chapter 4: (4.1 to 4.4, 4.6)

Unit III : (From Text Book – 1)

Chapter 5: (5.1 to 5.8)

Unit IV : (From Text Book – 2)

Chapter 1: (1.2.16 – 1.2.21)

Chapter 2: 2.1. 2.2,2.3(upto 2.3.3)

Unit V : (From Text Book – 2)

Chapter 2: (2.3.4 -2.3.14, 2.4)

Reference Books:

Reference Book - 1

Title of the book : Fuzzy set Theory and its applications

Name of the Author : H.J.Zimmer Mann

Publisher

Edition/Year

: Springer International Ltd

: Fourth Edition, 2006


Reference Book - 2

Title of the book : Fuzzy Commutative Algebra

Name of the Author : John .N Mordeson and T.S.Malik

Publisher : World Scientific Publishing Com.Pvt. Ltd

Edition/Year : 1998

Prepared by : Dr.S.Murugesan

Signature :




SATTUR – 626 203.



SYLLABUS


Semester : III No. of Hours allotted : 6 / Week

Paper : Core - Paper XI No. of Credits : 5

Title of the Paper-GALOIS THEORY AND LATTICES

Objectives:

To make them understand the aspects of field theory.

To know more concepts in Extension fields.

Unit I

Extension fields, the transcendence of e.

Unit II

Roots of polynomials, Construction with straightedge and compass, More about roots.

Unit III The elements of Galois Theory. Solvability by radicals.

Unit IV

Galois groups over the rationals, Finite fields.

Unit V Lattices

Lattices and Posets, Lattices as Posets, Lattices and Semilattices, Sublattices, Direct

Products, Distributive lattices, Modular and Geometric lattices.


Text Book 1

Titleof the book

: Topics in Algebra

Name of the authors : I. N. Herstein

Publisher : John Wiley and Sons

Edition/Year : Second Edition 1999

Text Book 2

Title of the book : Modern AppliedAlgebra

Name of the authors : Garret Birkhoff &H Thomas C. Bartee

Publisher : C. B. S.

Edition/Year :

Unit I : (From Text Book 1)


Unit II :


Unit III :


Unit IV :

Chapter 5: Sections 5.8 and Chapter 7, Section 7.1.

Unit V : (From Text Book 2) Chapter 9 ( Sections 9.1 to 9.6 )

Reference Books:

Reference Book - 1

Title of the book : A first Course in Algebra

Name of the Author : J.B. Fraleigh

Publisher : Addition –Wiely Longman Znc.Reading, Massachuetts

Edition/Year : 1999

Reference Book - 2

Title of the book : Basic Abstract Algebra

Name of the Author : P.B. Bhattacharya S K Jain, S.R Napul

Publisher :

Edition/Year :


Second Edition, 1995,(Reprinted 2009)

Prepared by : Dr. K. Nagarajan

Signature :



SATTUR – 626 203.



SYLLABUS

Programme : M. Sc., Mathematics Subject Code : P16MAC32

Semester : III No.of Hours allotted : 6 / Week

Paper : Core - Paper XII No.of Credits : 5

Title of the Paper MEASURE AND INTEGRATION

Objectives:

To give the comprehensive idea about the underlying principles of Lebesgue measure.

To describe the properties of Lebesgue measure.

Unit I

Lebesgue Outer Measure - Measurable Sets-Regularity .

Unit II

Measurable Functions - Borel and Lebesgue Measurability.

Unit III

Integration of Non-negative Functions – The general integral – Integration of series.

Unit IV

Riemann and Lebesgue integrals – The Four Derivatives – Continuous Non -

Differentiable Functions.

Unit V

Functions of Bounded Variations – Lebesgue Differentiation Theorem –

Differentiation and integration – The Lebesgue Set.


Text Book:

Title of the book

: Measure Theory and Integration

Name of the author : G. de Barra

Publisher

Edition/Year

Unit I :

: New Age International (P) Limited, Publishers

( formerly Willey Eastern Ltd)

: Reprint 2008.

Chapter 2: Sections 2.1 , 2.2 and 2.3.

Unit II :


Unit III :

Chapter 3: Sections 3.1, 3.2 and 3.3.

Unit IV :

Chapter 3: Sections 3.4,


Unit V :


Reference Book:

Reference Book - 1

Title of the book : Real Analysis

Name of the Author : H.L.Royden

Publisher

Edition/Year

Reference Book - 2

Title of the book

: MacMillan, New York

: Third Edition, 1988

: Measure Theroy

Name of the Author : Paul RHalmos

Publisher

Edition/Year

: Naros Publishing House

: Springer International Student Edition /1981


Signature :



SATTUR – 626 203.



SYLLABUS



Paper : Core - Paper XIII No.of Credits : 5

Title of the Paper -TOPOLOGY Objectives:

To lay the foundations for future study in analysis, in geometry and in algebraic

topology.

To develop the firm footing on the core subject of topology.

Unit I

Topological Spaces – Basics for a topology – The order topology – The product

topology on X x Y – The subspace topology – Closed sets and limits points – Continuous

functions – The product topology.

Unit II

The metric topology – Connected spaces – Connected subspaces of the real line.

Unit III

Compact spaces – Compact subspaces of the real line – Limit Point compactness –

Local Compactness.

Unit IV

Countability axioms – The separation axioms – Normal spaces.

Unit V

The Uryshon lemma – Completely regular - The Urysohn metrization Theorem – -

Imbedding Theroem – The Tietze Extension Theorem. .


Text Book:

Title of the book

: Topology

Name of the author : James R.Munkres

Publisher

Edition/Year

Unit I :

: Prentice – Hall of India, Private Ltd, New Delhi


Chapter 2: Sections 12 – 19

Unit II :

Chapter 2: Sections 20, 21.


Unit III:


Unit IV :

Chapter 4: Section 30, 31, 32.

Unit V:

Chapter 4: Sections 33,34, 35.

Reference Book:

Reference Book - 1

Title of the book : Introduction to Topology and Modern Analysis

Name of the Author : G.F. Simmons

Publisher

Edition/Year

Reference Book - 2

Title of the book

: Tata MacGrow Hill

: 2008

: Toplogy

Name of the Author : N James Dugundj

Publisher

Edition/Year

: Universal Book Stall, New Delhi

:1990

Prepared by : Dr. B. Meeradevi

Signature :




SATTUR – 626 203.


SYLLABUS



Paper : Core - Paper XIV No.of Credits : 4

Title of the Paper– STOCHASTIC PROCESSES

Objectives:

T o understand the concepts of Stochastic processes

Develop skills to know the nature of states of Markov Models.

Unit I

GENERATING FUNCTIONS AND MARKOV CHAINS

Generating Functions – Probability Generating Functions : Mean and Variance of

Bernoulli, Poisson, Geometric and Logarithmic distributions - Solutions to difference equations

using generating functions and method of characteristic functions - Definitions and Examples of

Stochastic processes - Markov Chains : Definitions and Examples – Transition Matrix –

Probability Distribution - Order of a Markov Chain – Markov chains as graphs - Higher Transition

Probabilities.

Unit II

STATES AND STABILITY OF MARKOV CHAIN

Classification of States and Markov Chain : Transient and persistent states - Determination of higher

transition probabilities - Stability of a Markov System , Ergodic theorem – Graph theoretic approach.

Unit III

POISSON PROCESS

Poisson Process and its Extensions: Poisson Process – Properties of Poisson process – Poisson process

and related distributions –Generalization of Poisson Process – Poisson cluster process - Pure birth

process: Yule furry process –birth immigration process- Time dependent Poisson Process


Unit-IV BIRTH DEATH PROCESS AND RENEWAL PROCESS

Birth-Death Process - Renewal Processes in Continuous Time –Renewal equation -

Stopping time - Wald‟s Equation – Elementary renewal theorems –Central limit theorem

on Renewals

Unit-V WIENER PROCESS

Markov Processes with Continuous State Space: Introduction –Brownian Motion – Wiener Process-

Differential equations for a Wiener Process - Kolmogorov equations – First Passage time distribution

for Wiener Process- Ornstein Uhlenbeck Process.

Text Book:

Title of the book : Stochastic Processes

Name of the author : J.MEDHI

Publisher

Edition/Year

Unit 1:

: New Age International Publishers

: Third Edition, 2009

Unit 2:

Unit 3:

Unit 4:

Unit 5:

Chapter 1: Sections 1.1.1, 1.1.2 , 1.5, Appendix: pp456-461




Chapter 3: Sections 3.4

Chapter 6: Sections 6.2 to 6.5.1, 6.5.5


Reference Book:

Reference Book - 1

Title of the book

: Introduction to Stochastic Processes

Name of the Author : Paul G. Hoel, Sidney C. Port, Charles J. Stone,

Publisher :

Edition/Year :


Reference Book - 2

Title of the book : Haughton Mifflin Comp.,

Name of the Author : A First Course in Stochastic Processes

Publisher : Samuel Karlin and Howard M.Taylor Academic Press

Edition / Year : 1972


Signature :



SATTUR – 626 203.


(For the candidates admitted from the year 2016-2017 onwards)

SYLLABUS

Programme : M. Sc., Mathematics Subject Code : P16MAE31 Semester : III No. of Hours allotted : 6 / Week

Part : Paper XV – Elective III(a) Credits :4

Title of the Paper-NUMERICAL ANALYSIS

Objectives:

To provide the student an understanding of the basic principles of numerical

methods and to apply them in solving algebraic equations and ordinary differential

equations numerically.

To introduce various difference operators to enable the students to apply them in

Interpolation and numerical differentiation and integration.

Unit I :

Introduction - Bisection method – Method of False Position - Iteration Method – Newton-

Raphson Method – Ramanujan‟s Method

Unit II :

Secant Method–Muller‟s Method–Graffe‟s Root-Squaring Method–Lin-Bairstow‟s Method–

Quotient-Difference Method–Solution to Systems of Nonlinear Equations.

Unit III:

Introduction - Numerical Differentiation - Maximum and Minimum Values of a

Tabulated Function-Numerical Integration-Euler –Maclaurin Formula – Numerical Integration

with Different Step Sizes.

Unit IV :

Numerical Solution of Ordinary Differential Equations – Introduction – Solution by Taylor‟s

Series – Picard‟s Method of Successive Approximations – Euler‟s Method–Runge–Kutta

Maethods –Predictor-Corrector Methods.

Unit V:

Numerical Solution of Partial Differential Equations – Introduction –Laplace‟s Equation –

Finite-difference Approximations to Derivatives–Solution of Laplace‟s Equation.


Text Book:

Title of the book : Introductory Methods of Numerical Analysis

Name(s) of the Author(s): S.S. Sastry

Publisher : PHI Learning Private Limited

Edition/Year : Fifth Edition, 2012.

Unit I


Unit II


Unit III


Unit IV


Unit V


Reference Books:

Reference Book: 1.

Title of the book : Applied Numerical Analysis,

Name(s) of the Author(s): C.F.Gerald and P.O.Wheatley,

Publisher : Addison Wesley,

Edition/Year : Fifth Edition, 2008

Reference Book – 2 ,

Title of the book : Elementary Numerical Analysis.

Name(s) of the Author(s): Samuel D Conte and Carl de Boor,

Publisher : Tata MacGraw Hill Pvt.Ltd,

Edition/Year : Third Edition, 1980.

Prepared by : Mr. D. K. Nathan

Signature :


SRI S.RAMASAMY NAIDU MEMORIALCOLLEGE (An Autonomous Institution Re-accredited with „A‟ Grade by NAAC)



SYLLABUS

Programme : M. Sc., Mathematics Subject Code : P16MAE32


Paper : Paper XV - Elective III(b) No.of Credits : 4

Title of the Paper-INTEGRAL EQUATIONS

Objectives:

To provide the basic knowledge of Integral Equations.

To introduce various difference applications of Integral Equations.

Unit I

Integral Equation – Differentiation of a function Under an Integral Sign- Relation between

Differential and Integral Equations- Illustrative Examples.

Unit II

Solution of Nonhomogeneous Volterra‟s Intergral Equation of Second kind by the Method of

Successive Substitution- Solution of Non-homogeneous Volterra‟s Intergral Equation of Second

Kind by the Method of Successive Approximation-Determination of Some Resolvent Kernels-

Volterra Intergral Equation of the First Kind-Solution of the Fredholm Intergral Equation by the

Method of Successive Substitutions- Iterated Kernels-Solution of the Fredholm Intergral Equation

by the Method of Successive Approximation- Reciprocal Functions- Volterra Soluctions of

Fredholm‟s Equations.

Unit III

Fredholm First Theorem-prove that the solution-Every Zero of Fredholm Function is a pole of

the Resolvent Kernel-If a real Kernel has a Complex Eigen Value then it also Contains the

Conjugate Eigen Value to -Hadamard‟s Theorem-Convergence proof-Fredholm second-Theorem-

Fredholm‟s Assiciated Equation. Characteristic Solutions-Fredholm‟s Third Theorem.

Unit IV

All Iterated Kernels of a Symmetric Kernel are also Symmetric-Orthogonality-Orthogonality of

Fundamental Functions –Eigen Value of Symmetric Kernel are Real.Real Charateristic Constants-

Expansion of a Symmetric Kernel in Eigen Functions-Symmetric Kernels with a Finite Number of


Eigen Values-Symmetric Kernels –with a Finite Eigen Value - Sequence of the pth Power of the

Eigen Value of the Iterated Kernal-Fourier series of power of the Eigen Value of the Iterated

Kernal-Hillbert – Schmidt Theorem- The inequalities of Schwarz and Minkowski-Hilbert‟s

Theorem-Complete Normalized Orthogonal System of Fundamental Functions-Bessel Intequality-

Riesz –Fischer Theorem-Representation by a liner Combination of the Charateristic Functions-

Schidt‟s Solution of the Non-Homogeneous Integral Equation – Solution of the Non-Homogeneous

Integral Equation-Solution of the Fredholm Integral Equation of first Kind-

Unit V

Introduction – Initial Value Problem- Boundary Value Problem – Deformation of a Rod –

Determination of Periodic Solutions-Green‟s Function.Construction of Green‟s Function-Particular

Case-Influence Function Construction of Green‟s Function when the Boundary Value Problem

Contains a Parameter.

Text Book:

Title of the book

: Integral Equations

Name of the author : Shanti Swarup and Shiv Raj Sing

Publisher : Krishna Prakashan Media (P) Ltd. India

Edition / Year : 25th

Edition, 2015.

Unit I : .

Chapter 1: All Sections

Unit II :


Unit III :


Unit IV :


Unit V :

Chapter 5 : All Sections

Reference Books:

Reference Book - 1

Title of the book : Integral Equations

Name of the author : David Porter & David S.G. Stirling

Publisher

Edition/Year

: Cambridge University Press ,NewYork.

: First Edition 1990.



Title of the book : Integral Equations and Boundary Value Problems

Name of the author : Dr. M.D.Raisinghania

Publisher : S. CHAND & COMPANY PVT.LTD

Edition/Year : Second Edition, 2008


Signature :




SATTUR – 626 203.


SYLLABUS


Semester : IV No.of Hours allotted : 6 / Week

Paper : Core - Paper XVI No.of Credits : 5

Title of the Paper :COMPLEX ANALYSIS

Objectives:

To give a comprehensive idea about the underlying principles of Complex analysis.

To introduce the theory of analytic function, complex integration and bilinear

transformations.

Unit I: Complex Functions

Concept of Analytic Functions – Limits and Continuity – Analytic Functions –

Polynomials – Rational Functions – Elementary Theory of Power Series - Sequences -

Series – Uniform Convergence – Power series – Abel‟s Limit Theorem.

Unit II: Analytic Functions as Mappings

Conformality – Arcs and Closed Curves – Analytic Functions in Regions – Conformal

Mapping-Lengths and Arcs-Linear Tranformations – The Linear Group – The Cross Ratio -

Symmetry.

Unit III : Complex Integration

Line Integrals –Rectifiable Area – Line Integrals as Functions of Arcs – Cauchy‟s

Theorem for a Rectangle – Cauchy‟s Theorem in a Disk - The Index of a Point with Respect

to a Closed Curve – The Integral Formula - Higher Derivatives.

Unit IV : Local Properties of Analytical Functions

Removable Singularities – Taylor‟s Theorem – Zeros and Poles – The Local Mapping

– The Maximum Principle - Chains and Cycles - Simple Connectivity – Homology - The

General Cauchy‟s Theorem - The Residue Theorem – The Argument Principle – Evaluation

of Definite Integrals.


Unit V : Harmonic Functions, Series and Product Developments

Definition and Basic Properties – The Mean - Value Property – Poission‟s Formula –

Schwarz‟s Theorem – The Reflection Principle - Power Series – Expansions - Weierstrass‟s

Theorem – The Taylor Series – The Laurent Series.

Text Book:

Title of the book : Complex Analysis

Name of the author : V. Ahlfors

Publisher

Edition/Year

Unit I :

: MeGraw Hill ISE

: III Edition 1981


Unit II :

Chapter 3 : Sections 2, 3 ( 3.1 to 3.3 only) .

Unit III :


Unit IV :

Chapter 4: Sections 3, 4(4.1 to 4.5 only) and 5

Unit V :

Chapter 4 : Section 6

Chapter 5 : Section 1.

Reference Book:

Reference Book - 1

Title of the book : Complex Analysis

Name of the Author : V.Karunagaran ,

Publisher

Edition/Year

Reference Book - 2

Title of the book

: Narosa Publications,

: Second Edition,

: Introduction to Complex Analysis

Name of the Author : H.A Pirestley

Publisher

Edition/Year

: Oxford University Press,

: Second Edition /2006

Prepared by : Dr.S.Murugesan

Signature :




SATTUR – 626 203.


SYLLABUS



Paper : Core - Paper XVII No.of Credits : 5

Title of the Paper -FUNCTIONAL ANALYSIS

Objectives:

To enrich the students with the advanced topics of functional analysis.

To get the comprehensive idea about the core principles of Gelfand mappings.

Unit I

Normed Spaces - Continuity of Linear maps .

Unit II

Hahn-Banach Theorems – Banach spaces.

Unit III

Uniform Boundedness Principle – Closed Graph Theorem and Open Mapping

Theorem , Bounded inverse Theorem.

Unit IV

Inner Product spaces – Orthonomal sets – Projection and Riesz Represetation

Theroems.

Unit V

Bounded Operators and Adjoints – Normal, Unitary and Self-adjoint operators.

Text Books Title of the book : Functional Analysi

Name of the author: B.V.Limaye

Publisher

Edition/Year

: New Age international Ltd

: Second Edition 1996


Unit I

Unit II

Unit III

Unit IV

Unit V

:

:

:

:

:

Chapter II: Sections5: 5.1 to 5.7.

Section 6: 6.1 to 6.8

Chapter II : Sections 7 : 7.1 to 7.12

Section 8: 8.1 to 8.4

Chapter III: Sections 9 : 9.1 to 9.3(Pages 138 to 144 only)

Section 10: 10.1 to 10.7

Chapter VI: Sections 21: 21.1 to 21.3(b)

Section22: 22.1 to 22.9

Section 24: 24.1 to 24.06 (Pages 420 to 431 only).

Chapter VII: Section: 25: 25.1 to 25.5

Section 26: 26.1 to 26.5 (Pages 460 to 473)

Reference Book:

Reference Book - 1

Title of the book

: Functional analysis

Name of the Author : Walter Rudin

Publisher

Edition/Year

Reference Book - 2

Title of the book

: Mac Graw Hill international Student Limited

: 1976

: Foundation of Functional Analysis

Name of the Author : S.Poonusamy

Publisher

Edition/Year

: Narosa Publications

: 2006


Signature :




SATTUR – 626 203.


SYLLABUS



Paper : Core - Paper XVIII No.of Credits : 5

Title of the Paper -OPERATIONS RESEARCH

Objectives:

To develop an understanding of various OR tools and their applications to real life

problems.

To become familiar with various OR models.

Unit I : Network Models

Scope of Network Applications - Network Definitions - Minimal Spanning Tree

Algorithm - Shortest Route Problem -Maximal Flow Model - CPM and PERT

Unit II : Queuing Systems

Elements of Queuing Model - Role of Exponential Distribution - Pure Birth and Death

Models, Relationship Between Exponential and Poisson Distributions - Generalized Poisson

Queuing Model

Unit III : Specialized Poisson Queues

Specialized Poisson Queues - M/G/1: (GD/∞/∞) Pollaczek – Khintchine (P-K) Formula -

Other Queuing models - Queuing Decision Models

Unit IV : Classical Optimization Theory

Introduction -Unconstrained Problems - Constrained Problems

Unit V Non Linear Programming Algorithms

Unconstrained Nonlinear Algorithms - Constrained Algorithms

Text Book:

Title of the book

: Operations Research : An Introduction

Name of the author : H. A. Taha

Publisher

Edition/Year

: Prentilce-Hall of India Pvt.Ltd

: VI Edition, 1997


NOTE: Section C of the Question paper for the end semester examination will contain

only numerical problems.

Unit I :

Chapter 6, Sections 6.1 to 6.5, 6.7.

Unit II :

Chapter 17, Sections 17.1 to 17.5.

Unit III :


Unit IV :


Unit V :

Chapter 21, Sections 21.1 and 21.2.

Reference Books

Reference Books -1

Title of the book

Name of the Author

Publisher

Edition/Year

Reference Books -2

Title of the book

Name of the Author

Publisher

Edition/Year

: Linear Programming and Networks Flows

: Mokhtar S. Bazara et. al,

: John Wiley and sons, Singapore,

: 1990

: Principles of Operation Research with applications to

Managerial Decisions

: Harvey M. Wagner

: Pretice Hall of Private Ltd, New Delhi


Prepared by : Dr. V.Thiripurasundarai

Signature :


SRI S.RAMASAMY NAIDU MEMORIAL COLLEGE (An Autonomous Institution Re-accredited with ‘A’ Grade by NAAC)

SATTUR – 626 203.



SYLLABUS

Programme : M. Sc., Mathematics Subject Code : P16MAE41


Paper : Paper XIX - Elective IV (a) No.of Credits : 4

Title of the Paper- ADVANCED TOPOLOGY

Objectives:

To enable the students to know more about Topology.

To develop a firm footing on the elective Topology.

Unit I

The Tychonoff Theorem - The Stone-Cech Compactfication – local finiteness.

Unit II

The Nagata-Smirnov Metrization Theorems – Paracompactness – Stone‟s Theorem –

The Smirnov Metrization Theorem.

Unit III

Complete Metric Spaces – A Space Filling Curve.

Unit-IV :

Compactness in Metric Spaces – Pointwise and Compact convergence – Ascolo‟s

Theorem.

Unit-V:

Baire Spaces – A nowhere differentiable function.

Text Book:

Title of the book : TOPOLOGY A first course

Name of the author : James R. Munkres

Publisher : Prentice Hall of India Private Ltd, New Delhi

Edition/Year : Second Edition ,2011


Unit I

:

Chapter 5: Section 37, 38;

Chapter 6: Section 39

Unit II : Chapter 6: Sections 40 - 42.

Unit III

Unit IV

Unit V

:

:

:


Chapter 7: Sections 45 to 47


Reference Book:

Reference Books -1

Title of the book

: General Topology

Name of the Author : Stephen Willard

Publisher

Edition/Year

Reference Books -2

Title of the book

: Addison-Wesley Pub.Co.

: 1970

: Toplogy

Name of the Author : James Dugundj

Publisher

Edition/Year

: Universal Book Stall, New Delhi.

: 1990


Signature :



(An Autonomous Institution Re-accredited with ‘A’ Grade by NAAC)

SATTUR – 626 203.


SYLLABUS



Paper : Paper XIX – Elective IV(b) No.of Credits : 4

Title of the Paper- NUMBER THEORY AND CRYPTOGRAPHY

Objectives:

To enable the students to know more about numbers.

To provide an introduction to analytic number theory.

To introduce the recent topics of Cryptography with applications.

Unit I

Introduction, Divisibility, Greatest common divisor, Prime numbers, The fundamental

theorem of arithmetic, The series of reciprocals of the primes, The Euclidean algorithm, The

GCD of more than two numbers, The Mobius function µ(n), The Euler totient function φ(n),

A relation connecting φ and µ , A product formula for φ(n), The Dirichlet product of

arithmetical functions, Dirichlet inverses and the Mobius inversion formula , The Mangoldt

function (n).

Unit II

Multiplicative functions , Multiplicative functions and Dirichlet multiplication, The

inverse of a completely multiplicative fucntion. Liouville‟s function (n), The divisor

functions (n), Generalized convolutions Formal power series ,The Bell series of an

arithmetical function, Bell series and Dirichlet multiplication , Derivatives of arithmetical

functions, The Selberg identity.

Unit III

Definition and basic properties of congruences, Residue classes and complete

residue systems, Linear congruences, Reduced residue systems and the Euler-Fermat

theorem, Polynomial congruences modulo p, Langrange‟s theorem . Applications of

Lagrange‟s theorem. Simultaneous linear congruences , The Chinese remainder theorem.


Unit-IV : Cryptography

Some simple crptosystems – Enciphering matrices.

Unit-V: Public Key

The idea of public key cryptography – RSA – Discrete log(The index – calculus

algorithm is not included) – Knapsack.

Text Books:

Text Books -1

Title of the book : Introduction to Analytic Number Theory

Name of the author: T.M. Apostol

Publisher

Edition/Year

Text Books -2

: Narosa Publishing Ltd, India

: III edition, 1991

Title of the book : A Course in number theory and cryptography

Name of the author: Neal Koblitz

Publisher

Edition/Year

: Springer International Edition,

: Second Edition, Fourth Indian Reprint 2010.

Unit I

Unit II

Unit III

Unit IV

Unit V

: (From Text book – 1)






Chapter 5 : Sections 5.1 to 5.7.


Chapter III: Sections 1 and 2.


Chapter IV: Sections 1 to 4.


Reference Books:

Reference Books -1

Title of the book

: An Intoruction to Theory of Numbers

Name of the Author : Niven and Zuckermann

Publisher

Edition/Year

Reference Books -2

Title of the book

: Wiley Eastern Ltd.

: 13rd Edition / 1972

: Introduction to cryptographyy

Name of the Author : Neal Konlitz

Publisher

Edition/Year

: Chapman and Hall /CRC

: 2nd

Edition /2007


Signature




SATTUR – 626 203.


(Those who joined in 2016-2017 and after)

SYLLABUS

Programme : M.Sc.,Mathematics Subject Code :P16MAPT41

Semester : IV No.of Hours allotted :6/Week

Paper : Core-Paper XX No.of Credits :4

Title of the Paper-PROJECT

Objectives:

To develop the ability of the students to prepare a project.

To get clear idea about the new concepts in Mathematics apart from the syllabus.

Regulations for the Project Report

The topic of the project may be based on research articles from mathematical journals or

any topic not covered in the M. Sc syllabus.

Evaluation method for project

Max Marks

Internal External Credits

Project Report 40 40

Viva Voce 20

Total 100 4

1. Internal examiners are the respective supervisors.

2. Project Reports are evaluated by both internal and external Examiners

3. Viva Voce examination is conducted evaluated by the external examiner.

The format of the project report should have the following components.

First page should contain:

Title of the project report.

Name of the candidate

Register number.

Name of the supervisor.

Address of the institution

Month & year of submission.


Contents .

Declaration by candidate

Certificate by supervisor.

Acknowledgement

Preface

Chapter 1 - Preliminaries.

Other chapters.

References.

The report of the project must be in the prescribed form. It should be typed

neatly in MSword with the equation editor or using Latex. The font size of the letter

should be 12 or 13 points with double space.

The number of pages in the project may vary from 40 to 50 .

Each page should contain atleast 18 lines.

Four copies of the project report with binding should be submitted.

Prepared by:

Signature :




SATTUR – 626 203.


(Those who joined in 2016-2017 and after)

SYLLABUS

Programme : M. Sc., Mathematics Subject Code :P16MAX41

Semester : IV No. of Credits : 3

Extra Credit paper

Title of the Paper: Model Paper for NET/SET Examination

Objectives:

To create an awareness of the NET/SET Examination

To make the students prepared for NET / SET Examinations

UNIT-I

Algebra: Permutations- Combinations- Pigeon – hole principle – inclusion –exclusion

principle - derangements- Group – subgroups , normal subgroups- quotient group –

homeomorphisms- cyclic groups – permutation groups- cayley‟s theorem – Sylow‟s theorems

–Rings- Ideals – U.F.D – P.I.D- Euclidean Domain.

UNIT-II

Linear Algebra: Vector spaces- subspaces – Linear dependence – basis – dimension –

algebra of linear transformations – Algebra of matrices – rank and determinant of matrices –

linear equations – Eigen values & Eigen vectors – Cayley Hamilton theorem – Matrix

representation of Linear transformation, Change of basis – Canonical forms – diagonal forms

– triangular forms – Jordan forms – Inner product spaces – orthonormal basis – Quadratic

Forms .

UNIT-III

Analysis: Finite , countable and uncountable sets – Archimedean property – supremun-

infimum - Sequences and Series – liminf - limsup – Continuity – Uniform Continuity –

Differentiability – Mean value theorem – Sequence and series of functions – Uniform

convergence – Riemann Integral – improper Integral‟s – Functions of general valuables –

directional derivative- partial derivative – derivative, as a Linear transformations – Metric

spaces – compactness – connectedness – space of all continuous functions.


UNIT-IV

Complex Analysis : Analytic functions – C-R equations - Cantour Integral – Cauchy‟s

theorem – Cauchy‟s integral formula – Lioville‟s theorem – Maximum modulus principle –

Schwarz lemma – Open mappings – Mobius transformations – Taylor series- Calculus of

residues.

Topology: basis – dense sets – subspace and product topology – Separation axioms –

connectedness and compactness.

UNIT-V

Differential equations ODE : existence and uniqueness of solutions of initial value pbms

for first order ordinary differential equations – Singular solutions of first order ODE –

General theory of homogeneous and non homogeneous linear ODE – Variations of

parameters – Sturm – Lioville boundary value problem.

PDE :

Lagrange and Charpit methods for solving first order PDC – classification of second order

PDE- General solutions of higher order PDE, with constant coefficients.

Sample space, discrete probability, independent events, Bayes theorem. Random variables

and distribution functions (univariate and multivariate); expectation and moments.

Independent random variables, marginal and conditional distributions. Characteristic

functions. Probability inequalities (Tchebyshef, Markov, Jensen). Modes of convergence,

weak and strong laws of large numbers. Central Limit theorems (i.i.d.case).

Markov chains with finite and countable state space, classification of states, limiting

behaviour of n-step transition probabilities, stationary distribution.

Reference Books:

UNIT I: Topics in Algebra by I.N. Herstein WILEY Publication, second edition .

Conlemporary, Abstaract Algebra, Joseph A Gallian, Narosa Publicizing House,

Fourth Edition.


UNIT II: Linear Algebra byTenneth Hoffman RAY KUNZE, PHI Learning Private

Limited second edition. Linear Algebra, Stephen H.Friedbery, Arnold J.Insel,

Lawrence E.Spence,PHI Learning Private Limited Fourth edition

UNIT III: Principles of Mathematical Analysis by Walter Rudin, MOGRAW-HILL

Publication, third edition.

Real Analysis, N.L. Carothers, Cambridge University Press.

UNIT IV: Foundation of Complex Analysis by S. Ponnusamy, Narosa Publications, second

edition. Topology by James R. Munkers PHI learning private Limited, second

edition. Function of one complex variable, John. B. Conway, Narosa publishing

House, Second edition.

UNIT V: Differential equations by D. Raisinghania

An Introduction to Ordinary Differential Equations

A. Codington, PHI learning private Limited.

Prepared by:

Signature :

CHAIRMAN DEAN

sri s.ramasamy naidu memorial college...

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